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Topology/Geometry Seminar

Minimal surfaces from deformations of circle patterns

Dr. Wai-Yeung Lam: Brown University

Location:  Hill 005
Date & time: Tuesday, 21 February 2017 at 3:30PM - 3:31PM

William Thurston introduced circle packings to approximate holomorphic functions. Burt Rodin and Dennis Sullivan proved the convergence of the analogue of Riemann maps for circle packings. Oded Schramm further extended the idea by considering circle patterns, where circles are allowed to intercept with each other. We present a discrete analogue of the Weierstrass representation for minimal surfaces in terms of discrete holomorphic quadratic differentials. Given a triangle mesh in the plane, a circle pattern is induced by the circumscribed circles. We investigate infinitesimal deformations of the triangle mesh that preserve the intersection angles of the circumscribed circles. We then deduce discrete holomorphic quadratic differentials from the change in cross ratios, which yield polyhedral surfaces with vanishing mean curvature. A similar story is obtained if we replace the intersection angles by length cross ratios.

This is joint work with Ulrich Pinkall.

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Department of Mathematics

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